Symmetric Finsler Spaces Research Articles

Pseudo-Finsler Radially Symmetric Spaces

We introduce the concept of radially symmetric pseudo-Finsler spaces, which generalize the notion of symmetric Finsler spaces, and prove that this concept is equivalent to the preservation of flag curvature by parallel transport together with reversibility. As a consequence, reversible pseudo-Finsler manifolds with constant flag curvature are radially symmetric.

On the existence of bi-invariant Finsler metrics on Lie groups

In this paper, we study the geometry of Lie groups with bi-invariant Finsler metrics. We first show that every compact Lie group admits a bi-invariant Finsler metric. Then, we prove that every compact connected Lie group is a symmetric Finsler space with respect to the bi-invariant absolute homogeneous Finsler metric. Finally, we show that if G is a Lie group endowed with a bi-invariant Finsler metric, then, there exists a bi-invariant Riemanninan metric on G such that its Levi-Civita connection coincides the connection of F.

The Binet–Legendre Metric in Finsler Geometry

For every Finsler metric $F$ we associate a Riemannian metric $g_F$ (called the Binet-Legendre metric). The transformation $F \mapsto g_F$ is $C^0$-stable and has good smoothness properties, in contrast to previous constructions. The Riemannian metric $g_F$ also behaves nicely under conformal or bilipshitz deformation of the Finsler metric $F$. These properties makes it a powerful tool in Finsler geometry and we illustrate that by solving a number of named Finslerian geometric problems. We also generalize and give new and shorter proofs of a number of known results. In particular we answer a question of M. Matsumoto about local conformal mapping between two Minkowski spaces, we describe all possible conformal self maps and all self similarities on a Finsler manifold. We also classify all compact conformally flat Finsler manifolds, solve a conjecture of S. Deng and Z. Hou on the Berwaldian character of locally symmetric Finsler spaces, and extend the classic result of H.C. Wang about the maximal dimension of the isometry groups of Finsler manifolds to manifolds of all dimensions. Most proofs in this paper go along the following scheme: using the correspondence $F \mapsto g_F$we reduce the Finslerian problem to a similar problem for the Binet-Legendre metric, which is easier and is already solved in most cases we consider. The solution of the Riemannian problem provides us with the additional information that helps to solve the initial Finslerian problem. Our methods apply even in the absence of the strong convexity assumption usually assumed in Finsler geometry. The smoothness hypothesis can also be replaced by that of partial smoothness, a notion we introduce in the paper. Our results apply therefore to a vast class of Finsler metrics not usually considered in the Finsler literature.

Locally Symmetric Homogeneous Finsler Spaces

Let $(M,F)$ be a connected Finsler space and $d$ the distance function of $(M,F)$. A Clifford translation is an isometry $\rho$ of $(M,F)$ of constant displacement, in other words such that $d(x,\rho(x))$ is a constant function on $M$. In this paper we consider a connected simply connected symmetric Finsler space and a discrete subgroup $\Gamma$ of the full group of isometries. We prove that the quotient manifold $(M, F)/\Gamma$ is a homogeneous Finsler space if and only if $\Gamma$ consists of Clifford translations of $(M,F)$. In the process of the proof of the main theorem, we classify all the Clifford translations of symmetric Finsler spaces.

On the classification of weakly symmetric Finsler spaces

In this paper, we give the classification of some special types of weakly symmetric Finsler spaces. We first present a general principle to classify weakly symmetric Finsler spaces and also give a method to figure out the Berwald spaces among the class of weakly symmetric Finsler spaces. Then we give an explicit classification of weakly symmetric Finsler spaces with reductive isometric groups as well as the left invariant weakly symmetric Finsler metrics on nilpotent Lie groups of the Heisenberg type. As an application, we obtain a large number of high-dimensional examples of reversible Finsler spaces which are non-Berwaldian and with vanishing S-curvature, a kind of spaces which are sought after in an open problem of Z. Shen.

WEAKLY SYMMETRIC FINSLER SPACES

In this paper, we introduce the notion of weakly symmetric Finsler spaces and study some geometrical properties of such spaces. In particular, we prove that each maximal geodesic in a weakly symmetric Finsler space is the orbit of a one-parameter subgroup of the full isometric group. This implies that each weakly symmetric Finsler space has vanishing S-curvature. As an application of these results, we prove that there exist reversible non-Berwaldian Finsler metrics on the 3-dimensional sphere with vanishing S-curvature. This solves an open problem raised by Z. Shen.

On generalized symmetric Finsler spaces

In this paper, we study generalized symmetric Finsler spaces. We first study some existence theorems, then we consider their geometric properties and prove that any such space can be written as a coset space of a Lie group with an invariant Finsler metric. Finally we show that each generalized symmetric Finsler space is of finite order and those of even order reduce to symmetric Finsler spaces and hence are Berwaldian.

An Algebraic Approach to Weakly Symmetric Finsler Spaces

AbstractIn this paper, we introduce a new algebraic notion, weakly symmetric Lie algebras, to give an algebraic description of an interesting class of homogeneous Riemann-Finsler spaces, weakly symmetric Finsler spaces. Using this new definition, we are able to give a classification of weakly symmetric Finsler spaces with dimensions 2 and 3. Finally, we show that all the non-Riemannian reversible weakly symmetric Finsler spaces we find are non-Berwaldian and with vanishing S-curvature. Thismeans that reversible non-Berwaldian Finsler spaces with vanishing S-curvaturemay exist at large. Hence the generalized volume comparison theorems due to Z. Shen are valid for a rather large class of Finsler spaces.

On weakly symmetric Finsler spaces

In this paper, we study weakly symmetric Finsler spaces. We first study an existence theorem of weakly symmetric Finsler spaces. Then we study some geometric properties of these spaces and prove that any such space can be written as a coset space of a Lie group with an invariant Finsler metric. Finally we show that each weakly symmetric Finsler space is of Berwald type.

On symmetric Finsler spaces

In this paper, we study symmetric Finsler spaces. We first study some geometric properties of globally symmetric Finsler spaces and prove that any such space can be written as a coset space of a Lie group with an invariant Finsler metric. Then we prove that a globally symmetric Finsler space is a Berwald space. As an application, we use the notion of Minkowski symmetric Lie algebras to give an algebraic description of symmetric Finsler spaces and obtain the formulas for flag curvature and Ricci scalar. Finally, some rigidity results of locally symmetric Finsler spaces related to the flag curvature are also given.

ON PROJECTIVE MOTION IN FINSLER SPACES II

K. Takano [4] in a series of papers has studied and developed affine motion in non-Riemannian K*-spaces. R.S. Sinha [6] has studied affine motions in recurrent Finsler spaces. The existence of projective motion in a symmetric Finsler space was discussed by F.Meher [7].Let ξi be a contravariant vector generating an infinitesimal mapping n: Fn → F¯ n of type x¯ i = xi + ξ i(x)dt. Such a mapping, when preserving the parallelism of a pair of any two vectors in Fn, defines an affine motion. The above infinitesimal transformation defines a projective motion if it transforms the system of geodesics into that of geodesics. The present authors [8] have discussed projective curvature collineation by considering vanishing of the Lie-derivature of the curvature tensor in Finsler space. The purpose of this paper is to study projective motion in Finsler spaces for the curvature tensor R*ijkh. The notations used in the sequel are due to E. Cartan [1] and H. Rund [3].

PROJECTIVE CURVATURE COLLINEATION IN FINSLER SPACES

The present author (1984, 1987) has developed curvature collineation in Finsler spaces. Projective motion in a symmetric Finsler space was discussed by F.M. Meher (1972). An infinitesimal transformation is said to define a curvature collineation if the curvature tensor possesses vanishing Lie-derivative with respect to the transformation. The present paper is concerned with the study of curvature collineation in which infinitesimal transformation also admits projective motion in a Finsler space. The last section of this paper is devoted to discuss special cases where the vector field Vi(x) determines contra field and current field.

MOTION WITH CONTRA FIELD IN FINSLER SPACES

An affine motion of special type was developed in non-Riemannian K*-spaces by Takano [5]. He called such a motion an affine motion with contra field. The same concept was extended to Finsler spaces by Misra and Meher [1]. Later, the same authors [2] studied CA-motion in a projective symmetric Finsler space. An affine motion satisfying the condition υi (j) = ρδi j, where ρ(x, x) is a non-zero scalar function, is called affine motion with contra field or briefly CA-motion. Recently the present author [4] has discussed projective motion in Finsler space F n. The purpose of this paper is to study affine and projective motions with contra field in Finsler spaces. The notations of Rund [3] are used in the sequel.

Locally symmetric Finsler spaces in negative curvature

É. Cartan introduced in 1926 the Riemannian locally symmetric spaces, as the spaces whose curvature tensor is parallel. They also owe their name to the fact that, for each point, the geodesic reflexion is a local isometry. The aim of this Note is to announce a strong rigidity result for Finsler spaces. Namely, we show that a negatively curved locally symmetric (in the first sense above) Finsler space is isometric to a Riemann locally symmetric space.

The general four-dimensional spherically symmetric Finsler space

Finsler geometries give natural generalisations of Riemannian geometries, and hence possible natural extensions of general relativity. In this latter (gravitational) context, it is of particular interest to find the general spherically symmetric Finsler metric on ℝ4. In this paper, we derive this metric. The general solution is given in two alternative forms, the second of which allows easy comparison with Riemannian-like metrics. We also derive the general axially symmetric Finsler metric on ℝ4.